# Show that $xy/z$ is constant given conditions on $xy$ and $y/z$.

If $$xy$$ is constant whenever $$z$$ is constant,
and $$y/z$$ is constant whenever $$x$$ is constant, then show that $$xy/z$$ is constant.

My work: Write $$xy = a$$, $$y/z = b$$. Then

$$xy^2/z = ab, \ \ \ xz = a/b.$$

Any help?

Remark: This shows up in physics when I'm trying to derive the ideal gas equation from Boyle and Charles' laws ($$PV =$$ constant and $$V/T =$$ constant).

• $xy/z=ab/y$, therefore the assertion will be true if and only if $y$ is constant (or there are some silly zeros here and there), and that's all there is to it.
– user239203
Sep 12, 2020 at 9:27
• @Gae.S. I see... the textbook mentioned z = constant for first equation and x = constant for second equation. I'll add this to the question... Thank you! Sep 12, 2020 at 9:29
• @Gae.S. updated the q. please see again Sep 12, 2020 at 9:30

## 4 Answers

My interpretation is that $$x, y, z$$ are independent, but happens that $$xy$$ depends only on $$z$$ and $$y/z$$ depends only on $$x$$. Then we write

$$\tag{1} xy = f(z), y/z = g(x)$$ for some functions $$f, g$$.

A priori, $$xy/z$$ is a function of $$3$$ variables and we want to show that it is constant. Since

$$f(z)/z =\frac{xy}{z} = x g(x),$$

One sees that the expression is independent of $$x, y$$ and $$y, z$$. Thus it is independent of $$x, y, z$$ and thus is a constant.

• Looks neat! so xg(x) is also constant, but how did you conclude it was also a constant? Sep 12, 2020 at 9:58
• I think I know how you concluded: zg(x) = f(z)/x this implies f(z) = zxg(x). so xg(x) has to be constant for f(z) is a function of z alone. Did I get this right? Sep 12, 2020 at 10:02
• Ahh ignore my comments. I see now. f(z)/z is clearly independent of x,y. And xg(x) is clearly independent of y,z. Brilliant!!!! Sep 12, 2020 at 10:06

If $$y=z=\frac1x$$, then $$xy$$ and $$\frac yz$$ are constants. However,$$\frac{xy}z=x,$$which will not be constant unless $$x$$ itself is constant.

If $$xy$$ is constant and $$z$$ is constant, then $$\frac{xy}{z}$$ must also be a constant.

This is because if $$xy = p$$ and $$z = q$$ where $$p, q$$ are constant, $$\frac{xy}{z} = \frac{p}{q}$$, and $$\frac{p}{q}$$ is constant since $$p, q$$ are constant.

If $$\dfrac {xy}{z}$$ is constant and $$xy$$ is also constant then $$z$$ is constant.

If in addition $$\dfrac{y}{z}$$ is also constant, $$y,$$ and so $$(x,y,z)$$ are all constants.

Note from Physics about ideal gas Law proportionalities not exactly.

When T is constant pressure P is proportional to 1/V ( Boyle), and when volume V is constant P is proportional to temperature T (Gay-Lussac) and when P is constant V is proportional to T (Charles).

When combined $$\dfrac{PV}{T}$$ is a new constant.